Proving 1=2 Bob Ross Style

hi I’m Toby and welcome to another
episode of the joys of mathematics if this is your first time watching then
here’s your chance to go and grab a notepad and pen so you can work along at
home and if you are a returned viewer thanks for coming back now today we’re
going to work through something that’s a little more abstract in its creation and
we’re going to do this to hopefully see that sometimes you can’t always trust
something on its first impression sometimes you need to look a little bit
closer because often things are not as they seem
so keep in mind that this piece today is going to be more of an abstract artwork
than a conventional one so I’ve got my trusty titanium white color here and
let’s get started the first thing that we’re going to do
is we’re going to let a equal b. a and b are our first two characters and they
are variables they could be anything that they ever wanted to be they have
the whole world as their oyster but today a and b have chosen to be exactly
the same thing now as we go along don’t be afraid of the mathematics here I’m
here with you and I’m going to help you go through it well what we’re going to
do is times both sides by b that will give us a times b is equal to b times
itself b squared so we’ve added a couple of happy little b’s here to this line we
added these to both sides so that’s fine we can add or multiply whatever we want
as long as we do it on each side same with division and subtraction there’s a
moral of the story here only do to others what you want done to yourself
and when you’ve got two sides of an equation anything you do to one side you
have to do to the other now let’s add some more happy little b’s and in fact
we’re going to add another b squared to each side of this equation all right got a lot of happy little bs
let’s just simplify this b squared plus b squared
could just be written as 2 b squared and this line for the side says the same now
you might be sort of wondering how this is going to end up anywhere but just
trust me sometimes you need to just go where the wind blows you and it is
pretty windy out here in the garden today I hope that it doesn’t cause too
many issues but let’s just follow along for the drift of the wind here and you
might have to trust me a little bit okay we’ve added b squared what if we do
we’ll got an ab what if we subtract the same term from both sides and that term
is going to be 2ab all right let’s do it to both sides this term that I
subtracted from both sides could have been literally anything since I did it
to both sides and it’s going to make the result the same but up until now we
haven’t done anything illegal with our algebra this is all this is all fine
okay so if we have one ab and we’re taking away 2 ab’s how many ab’s do
we have left well the answer is we’ll still have our b squared and minus one ab and on the side we’ve got to be squared minus 2 ab all right what I
want to do next is to factorize this line and I’m going to continue up here
to do that so there’s one common term to both sides and that is b squared minus ab on this side we could take a 1 out of the bracket if we really wanted to
somewhat unnecessary but we have 1 times B squared minus a B and that is equal to
well we can take a 2 outside the brackets now 2
times b-squared minus a B getting a little close to the edge there but I
think we’re alright okay so now we’ve got this line still nothing illegal has
happened you want to check my algebra you can go through there but now we have
something all right we have B squared minus a B on both sides of the equation
what we may want to do is to recognize that if we divide both sides
by the same thing that’s alright like we can divide both sides by B squared minus
a B and that would be the same as cancelling out this term if we’d
cancelled out this term in brackets then what we have left with is that one is
equal to two now here’s where it appears to be pretty abstract and in fact
something doesn’t quite add up here I’m going to put a little box of shame
around this because this means we’ve done something weird or something
illegal a long way we can’t just suddenly have one being equal to two
because that would fundamentally change so many things about mathematics we rely
on two being exactly double one and not being equal to each other so have a
little look at the algebra and see if you can identify where we went wrong and
why don’t worry if you can’t get it I’m going to show you in just a second but
if you’ve formed your opinion and then let’s have a look I stand by that all of
this algebra is correct the issue comes in at this line here now
we divided both sides by B squared minus a B and that would be fine as long as B
squared minus a B the stuff in brackets was not equal to zero if this stuff was
equal to zero this would amount to dividing both sides by zero and as you
might know dividing by zero is kind of forbidden and mathematics the answer to
that is undefined when you start to viding by zero things in the universe
stop making much sense you could already sort of argue that the
world doesn’t make a lot of sense to us you know good things happen to bad
people all the time and there’s a lot of things that are unexplained but right
here this would give us some weird things dividing by zero leads to all
sorts of things such as that one is equal to two and frankly this is
nonsense so we need to sort this out so let’s have a look B squared since a and
B are equal to each other is actually the same as a times B so this in
brackets is in fact equal to zero and what we have on this line up here is
that one times zero is equal to two times zero now I mean sure anything
times zero is going to be equal to zero so these 1 and two could have been
replaced by literally anything all this really shows is that zero is equal to
zero so if we’ve proved anything that’s what we’ve proved here not that one is
equal to two so there is a bit of a moral here and it was to not trust your
first impressions all the time you know when we did this step it might have
looked like it was legal and this might have looked like a legal result but we
used our sort of critical thinking we realized it probably doesn’t make sense
and there’s probably a reason why so the rule for mathematics is that whenever
you’re going to divide by something check that that something is not zero
but also in life you know sometimes things are too good or too weird to be
true so always look a little closer and don’t just take things at face value now
I’m going to rub this off here and make room for a little message a duster can
get pretty chalky so you just beat the devil out of it this episode of the joys
of mathematics is brough to you Skillshare Skillshare are an online learning
community with classes in things like business design and technology I’m
currently working through some courses on writing short stories and editing
videos in final cut if you want to learn new skills then as a special offer
for my viewers first 500 people to sign up for
Skillshare at the link in the description will get two months of
unlimited access for free that’s access to over 20,000 courses so I’m sure you
can find one there that you like this is also your personal invitation to
subscribe to my channel for more videos like this don’t forget to take a closer
look at the world around you and from the bottom of my heart I hope that you
have an absolutely mathematical day here’s some bonus after the credits
footage but I just wanted to show you guys something in the garden it’s over
here and it’s a moth or a butterfly perched on a little branch there now
that’s pretty nice but don’t look too close past him well you might see what
scared me earlier it seems to be a spider I’ll zoom in because I don’t want
to get too close it’s either the spider or like the skin of the spider but to be
honest I didn’t know that spiders head skin so I’m getting out of here


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